Remark on compressible Navier–Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier–Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier–Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.